
In this toy model, the fourth-order symmetry energy is totally determined by the symmetry energy and the EOS of SNM via the parameters \(\beta _{\mathrm {IS}}, ~\lambda _{\mathrm {IS}}\)  and the parameter \(\mu \) .
See Fig. REF  for the density dependence of the \(E_{\mathrm {sym},4}(\rho )\)  (left) and that of the ratio \(E_{\mathrm {sym,4}}(\rho )/E_{\mathrm {sym}}(\rho )\)  (right), adopting different values of \(L\)  while fixing the \(E_{\mathrm {sym}}(\rho _0)\)  at 32 MeV.
We have found that the fourth-order symmetry energy \(E_{\mathrm {sym,4}}(\rho _0)\)  at saturation density is generally \(\lesssim 2\,\mathrm {MeV}\)  (indicated by the red “+”). This value is also consistent with available empirical findings [1]}, [2]}, [3]}, [4]}.
